| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 1.1-a1 |
1.1-a |
$6$ |
$147$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$60.19571$ |
$\textsf{none}$ |
0 |
$\Z/7\Z$ |
$\textsf{potential}$ |
$-147$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7B.1.1[3] |
$1$ |
\( 1 \) |
$1$ |
$22162.25980$ |
0.671415 |
\( -7604567359488000 a^{5} - 7604567359488000 a^{4} + 38022836797440000 a^{3} + 30418269437952000 a^{2} - 45627404156928000 a - 28831103815680000 \) |
\( \bigl[0\) , \( -a^{4} + 3 a^{2} - a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( 171 a^{5} + 81 a^{4} - 934 a^{3} - 192 a^{2} + 1093 a - 400\) , \( -1717 a^{5} - 882 a^{4} + 9032 a^{3} + 2384 a^{2} - 9721 a + 2928\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(-a^{4}+3a^{2}-a-1\right){x}^{2}+\left(171a^{5}+81a^{4}-934a^{3}-192a^{2}+1093a-400\right){x}-1717a^{5}-882a^{4}+9032a^{3}+2384a^{2}-9721a+2928$ |
| 1.1-a2 |
1.1-a |
$6$ |
$147$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$60.19571$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-147$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7B.1.6[3] |
$2401$ |
\( 1 \) |
$1$ |
$0.188376100$ |
0.671415 |
\( -7604567359488000 a^{5} - 7604567359488000 a^{4} + 38022836797440000 a^{3} + 30418269437952000 a^{2} - 45627404156928000 a - 28831103815680000 \) |
\( \bigl[0\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( 681 a^{5} - 1179 a^{4} - 4774 a^{3} + 5988 a^{2} + 7513 a - 6850\) , \( 23972 a^{5} - 32308 a^{4} - 159295 a^{3} + 171151 a^{2} + 242140 a - 203233\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(681a^{5}-1179a^{4}-4774a^{3}+5988a^{2}+7513a-6850\right){x}+23972a^{5}-32308a^{4}-159295a^{3}+171151a^{2}+242140a-203233$ |
| 1.1-a3 |
1.1-a |
$6$ |
$147$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$60.19571$ |
$\textsf{none}$ |
0 |
$\Z/7\Z$ |
$\textsf{potential}$ |
$-147$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7B.1.1[3] |
$1$ |
\( 1 \) |
$1$ |
$22162.25980$ |
0.671415 |
\( 7604567359488000 a^{5} + 7604567359488000 a^{4} - 38022836797440000 a^{3} - 30418269437952000 a^{2} + 45627404156928000 a - 6017401737216000 \) |
\( \bigl[0\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( 131 a^{5} - 79 a^{4} - 814 a^{3} + 488 a^{2} + 1133 a - 800\) , \( -1218 a^{5} + 802 a^{4} + 7465 a^{3} - 4959 a^{2} - 10420 a + 7417\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(131a^{5}-79a^{4}-814a^{3}+488a^{2}+1133a-800\right){x}-1218a^{5}+802a^{4}+7465a^{3}-4959a^{2}-10420a+7417$ |
| 1.1-a4 |
1.1-a |
$6$ |
$147$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$60.19571$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-147$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7B.1.6[3] |
$2401$ |
\( 1 \) |
$1$ |
$0.188376100$ |
0.671415 |
\( 7604567359488000 a^{5} + 7604567359488000 a^{4} - 38022836797440000 a^{3} - 30418269437952000 a^{2} + 45627404156928000 a - 6017401737216000 \) |
\( \bigl[0\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 9 a + 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( -188 a^{5} + 589 a^{4} + 447 a^{3} - 2545 a^{2} + 1720 a - 203\) , \( -6938 a^{5} + 17357 a^{4} + 18336 a^{3} - 72391 a^{2} + 45500 a - 5253\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-9a+1\right){x}^{2}+\left(-188a^{5}+589a^{4}+447a^{3}-2545a^{2}+1720a-203\right){x}-6938a^{5}+17357a^{4}+18336a^{3}-72391a^{2}+45500a-5253$ |
| 1.1-a5 |
1.1-a |
$6$ |
$147$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$60.19571$ |
$\textsf{none}$ |
0 |
$\Z/7\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7Cs.1.1[3] |
$1$ |
\( 1 \) |
$1$ |
$22162.25980$ |
0.671415 |
\( 0 \) |
\( \bigl[0\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 3 a\) , \( a^{5} - 6 a^{3} + 2 a^{2} + 9 a - 6\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+3a\right){x}+a^{5}-6a^{3}+2a^{2}+9a-6$ |
| 1.1-a6 |
1.1-a |
$6$ |
$147$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$60.19571$ |
$\textsf{none}$ |
0 |
$\Z/7\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7Cs.1.1[3] |
$1$ |
\( 1 \) |
$1$ |
$22162.25980$ |
0.671415 |
\( 0 \) |
\( \bigl[0\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 9 a + 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( 2 a^{5} - a^{4} - 13 a^{3} + 5 a^{2} + 20 a - 3\) , \( -2 a^{5} + a^{4} + 11 a^{3} - 5 a^{2} - 13 a + 2\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-9a+1\right){x}^{2}+\left(2a^{5}-a^{4}-13a^{3}+5a^{2}+20a-3\right){x}-2a^{5}+a^{4}+11a^{3}-5a^{2}-13a+2$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
